where \(i\)  and \(j\)  take values \(1,2,3\) , the magnitude of the squared momentum is \(p^2=\sum _{j=1}^{3}\,p_{j}p_{j}\) , and \(\alpha =\alpha _0/M_{Pl}c=\alpha _0l_{Pl}/\hbar \)  with \(\alpha _0\)  to be a dimensionless constant, sometimes assumed to be
\({\cal O}(1)\)Recently, in Ref. [1]}, using a GUP with only a quadratic term in momentum, a numerical value of the dimensionless GUP parameter \(\beta _0\)  (with \(\beta _0 \sim \alpha ^{2}_{0}\) )
was obtained, namely \(82\pi /5\) . Furthermore, in Ref. [2]} a similar analysis
with linear \(+\)  quadratic GUP yielded \(\alpha _{0}\)  to be proportional to powers of the dimensionless ratio (\(M_{Pl}\) /M)..
since GUP as formulated allows for a general \(\alpha _0\)  and one does not yet have a direct measurement of quantum gravity parameters, we leave it arbitrary. This in turn implies the existence of an intermediate length scale and imposes meaningful bounds on the quantum gravity parameters, which can in principle be measurable in future experiments [3]}, [4]}, [5]}, [6]}.
